The Hamiltonian constraint in Polymer Parametrized Field Theory

TL;DR

This paper advances the quantization of the Hamiltonian constraint in 2D Parameterized Field Theory using Loop Quantum Gravity techniques, revealing new insights into the constraint algebra and habitats for quantum states.

Contribution

It introduces a new habitat supporting a non-trivial representation of the density 2 constraint algebra, and analyzes the continuum limit of quantum constraints in polymer quantization.

Findings

Hamiltonian constraint acts only on vertices with some ambiguities

Physical states lie in the kernel of the Hamiltonian constraint with tailored holonomies

The commutator of Hamiltonian constraints converges to zero in the continuum limit

Abstract

Recently, a generally covariant reformulation of 2 dimensional flat spacetime free scalar field theory known as Parameterised Field Theory was quantized using Loop Quantum Gravity (LQG) type `polymer' representations. Physical states were constructed, without intermediate regularization structures, by averaging over the group of gauge transformations generated by the constraints, the constraint algebra being a Lie algebra. We consider classically equivalent combinations of these constraints corresponding to a diffeomorphism and a Hamiltonian constraint, which, as in gravity, define a Dirac algebra. Our treatment of the quantum constraints parallels that of LQG and obtains the following results, expected to be of use in the construction of the quantum dynamics of LQG:(i) the (triangulated) Hamiltonian constraint acts only on vertices, its construction involves some of the same ambiguities as in LQG and its action on diffeomorphism invariant states admits a continuum limit (ii)if the regulating holonomies are in representations tailored to the edge labels of the state, all previously obtained physical states lie in the kernel of the Hamiltonian constraint, (iii) the commutator of two (density weight 1) Hamiltonian constraints as well as the operator correspondent of their classical Poisson bracket converge to zero in the continuum limit defined by diffeomorphism invariant states, and vanish on the Lewandowski- Marolf (LM) habitat (iv) the rescaled density 2 Hamiltonian constraints and their commutator are ill defined on the LM habitat despite the well defined- ness of the operator correspondent of their classical Poisson bracket there (v) there is a new habitat which supports a non-trivial representation of the Poisson- Lie algebra of density 2 constraints